Abstract
We consider the three standard Lanchester models of warfare (aimed-fire, unaimed-fire and asymmetric) with heterogeneous (mixed) forces on both sides. We begin by reviewing the homogeneous models, and then construct conserved quantities for the mixed models with separable kill-rates and random target allocation, commenting on the nature and allocation of unit types. Next we consider a more general semi-dynamical target allocation, construct a conserved quantity for the aimed-fire model and prove that the optimal strategy is to annihilate opposing unit types in succession. Finally, we make some comments on the optimal initial allocation of costed unit types in response to this.
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I thank Simon Eveson and Jamie Wood for discussions.
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Appendix: derivation of conserved quantities
Appendix: derivation of conserved quantities
Suppose that we seek, for Lanchester equations (12) for aimed fire with mixed forces on both sides and random allocations, a quadratic conserved quantity of the form
Differentiating and substituting-in equations (12), one finds that for dQ/dt=0 we must have, for all i, j, α, β,
These equations are inconsistent for general r α i and b i α, but are soluble in the separable case (13), when they become
yielding (14). Analogous derivations of (16,18) are straightforward.
Now consider our semi-dynamical target allocation. Once again there is no quadratic conserved quantity for inseparable kill-rates. For separable kill-rates we simply note that the dynamical equations
lead to conservation of
for the Red and Blue contributions to Q′ have the same derivative,
The implication is that Red should concentrate fire in order of decreasing b i r i, Blue in order of r α b α.
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MacKay, N. Lanchester models for mixed forces with semi-dynamical target allocation. J Oper Res Soc 60, 1421–1427 (2009). https://doi.org/10.1057/jors.2008.97
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DOI: https://doi.org/10.1057/jors.2008.97